two consecutive terms of a GP are the 2nd ,4th and 7th terms of an A.P respectively.find the common ration of the GP
so the GP terms are:
a+d, a+3d, and a+6d
(a+3d)/(a+d) = (a+6d)/(a+3d)
(a+6d)(a+d) = (a+3d)^2
a^2 + 7ad + 6d^2 = a^2 + 6ad + 9d^2
ad = 3d^2
divide by d
a = 3d
common ratio :
r = (a+3d)/(a+d)
= 6d/(4d)
= 3/2
To find the common ratio (r) of the geometric progression (GP), we have to use the given information about the arithmetic progression (AP) formed by the 2nd, 4th, and 7th terms.
Let's denote the second term of the GP as a and the common ratio as r.
1st term of AP: a
2nd term of AP: a + 2d (since the common difference (d) of the AP is 2)
3rd term of AP: a + 4d (since the common difference (d) of the AP is 2)
4th term of AP: a + 6d (since the common difference (d) of the AP is 2)
5th term of AP: a + 8d (since the common difference (d) of the AP is 2)
6th term of AP: a + 10d (since the common difference (d) of the AP is 2)
7th term of AP: a + 12d (since the common difference (d) of the AP is 2)
Now, we can write the following equations based on the given information:
a + r = a + 2d -> Equation 1 (2nd term of GP is the 2nd term of AP)
a + 8r = a + 6d -> Equation 2 (4th term of GP is the 4th term of AP)
a + 64r = a + 12d -> Equation 3 (7th term of GP is the 7th term of AP)
To solve these equations, we can equate the corresponding sides:
r = 2d -> Equation 4 (from Equation 1, a terms cancel out)
8r = 6d -> Equation 5 (from Equation 2, a terms cancel out)
Now, let's solve Equation 4 and 5 to find the values of r and d:
Substituting Equation 4 into Equation 5:
8(2d) = 6d
16d = 6d
16d - 6d = 0
10d = 0
d = 0
Since d = 0, the common difference of the AP is 0. But a geometric sequence cannot have a common difference of 0, as it would result in all terms being the same.
Therefore, there is no valid common ratio for the given geometric progression (GP) based on the provided information.
To find the common ratio of the geometric progression (GP), we need to consider that the 2nd, 4th, and 7th terms of the geometric progression are also the 2nd, 4th, and 7th terms of an arithmetic progression (AP).
Let's start by assuming that the GP has the terms a, ar, ar^2, ar^3, ... and the AP has the terms a, a + d, a + 2d, a + 3d, ..., where a is the first term and d is the common difference.
Given that the 2nd term of the GP equals the 2nd term of the AP, we have ar = a + d. --- Equation (1)
Similarly, the 4th term of the GP equals the 4th term of the AP, so ar^2 = a + 2d. --- Equation (2)
Also, the 7th term of the GP equals the 7th term of the AP, so ar^3 = a + 3d. --- Equation (3)
We have three equations (Equation 1, Equation 2, and Equation 3) with three unknowns (a, r, and d). We can solve these equations to find the value of r, the common ratio of the GP.
First, let's simplify Equation 1 by dividing both sides by a: r = 1 + d/a.
Next, let's simplify Equation 2 by dividing both sides by a: r^2 = 1 + 2d/a.
Similarly, simplify Equation 3 by dividing both sides by a: r^3 = 1 + 3d/a.
Now, we can substitute the value of r^2 from Equation 2 into Equation 3: (1 + 2d/a) * r = 1 + 3d/a. Expanding this equation gives us:
r + 2d/a * r = 1 + 3d/a.
Multiply both sides by a to eliminate the denominator:
ar + 2dr = a + 3d.
Using Equation 1 (ar = a + d), we can substitute ar with a + d in the above equation:
a + d + 2dr = a + 3d.
Simplifying further:
2dr - 2d = 2d.
Dividing both sides by 2d:
r - 1 = 1.
Adding 1 to both sides:
r = 2.
Hence, the common ratio of the GP is 2.